We first consider a mortality rate in the model (1):

X = ц(Б)Х — mX

where parameter m > 0 becomes not negligible when ц(Б) takes small values. In addition, we consider an additional compartment Xd that represents the accumulation of dead cells:

Xd = 5mX,

where the parameter 5 Є (0,1) describes the part of non-viable cells that are not burst. We assume that the burst cells recycle part of the substrate that has been assimilated but not yet transformed. Then, the dynamics of the substrate concentration can be modified as follows:

S = —^y^-X + A(1 — S)mX,

where A > 0 is recycling conversion factor. It appears reasonable to assume that the factor A is smaller that the growth one:

In the following we assume that the growth function ц(-) and the yield coefficient Y of the classical Monod’s model are already known. Typically, they can be identified by measuring the initial growth slope on a series of experiments with viable biomass and different initial concentrations, mortality being considered to be negligible during the exponential growth. We aim at identifying the three parameters m, 5 and A, and on-line reconstructing the variables

X and Xd, based on on-line observations of the substrate concentration S and the total biomass B = X + Xd.

Without any loss of generality, we shall assume that the growth function ц(-) can be any function satisfying the following hypotheses.

Assumption A2. The function ц(-) is a smooth increasing function with у (0) = 0.

For sake of simplicity, we normalise several quantities, defining

s = S, x = X/Y, Xd = Xd/Y, a = (1 — S)m and k = XY.

Then, our model can be simply written as

S = — у (s) x + kax,

X = y(s)x — mx, (3)

X d = mx — ax,

s(0) = so > 0, xd(0) = 0 and x(0) = xo > 0 .

Our purpose is to reconstruct parameters m, a and k and state variable x(-) or xd(•), under the constraints m > a and k < 1, that are direct consequences of the definition of a and Assumption A1. Moreover, we shall assume that a priori bounds on the parameters are known

i. e.

(m, a, k) Є [m—, m+] x [a—,a+] x [k—, k+] . (4)